\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 21, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/21\hfil Mild solutions]
{Mild solutions for semilinear fractional differential equations}

\author[G.  M. Mophou, G. M. N'Gu\'er\'ekata\hfil EJDE-2009/21\hfilneg]
{Gis\`ele  M. Mophou, Gaston M. N'Gu\'er\'ekata}  % in alphabetical order

\address{Gis\`ele  M. Mophou \newline
Universit\'e des Antilles et de la Guadeloupe,
D\'epartement de Math\'ematiques et Informatique, Universit\'e
des Antilles et de La Guyane, Campus Fouillole 97159
Pointe-\`a-Pitre  Guadeloupe (FWI)}
\email{gmophou@univ-ag.fr}

\address{Gaston M. N'Gu\'er\'ekata \newline
 Department of Mathematics, Morgan State
University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA}
\email{Gaston.N'Guerekata@morgan.edu, nguerekata@aol.com}

\thanks{Submitted October 28, 2008. Published January 23, 2009.}
\subjclass[2000]{34K05, 34A12, 34A40}
\keywords{Fractional differential equation}

\begin{abstract}
 This paper concerns the existence of mild solutions for  fractional
 semilinear differential equation with non local  conditions
 in the $\alpha$-norm. We prove  existence and uniqueness,
 assuming  that the linear part generates  an analytic compact
 bounded semigroup, and  the nonlinear part is a  Lipschitz
 continuous function with respect to the fractional power norm
 of the linear part.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

During the past decades, fractional differential equations have
attracted many authors (see for instance
\cite{lak,lak1,lak2,gis1,gaston,gaston1,pod,wei,zhang} and
references therein). This, mostly because it efficiently describes
many phenomena arising in Engineering, Physics, Economy, and
Science.

 Our aim in this  paper is to discuss the existence and
the uniqueness of  the mild solution for fractional semilinear
differential equation with nonlocal conditions :
\begin{equation}\label{e1.1}
\begin{gathered}
D^q x(t)=-Ax(t)+f(t,x(t),Bx(t)),\quad t\in [0,T],\\
x(0)+g(x)=x_0
\end{gathered}
\end{equation}
where $T>0$, $0<q<1$, $-A$ generates an analytic compact semigroup
$(S(t))_{t\geq 0}$ of uniformly bounded linear operators on a
Banach space $\mathbb{X}$. The term $Bx(t)$ which may be
interpreted as a control on the system is defined by:
$$
Bx(t) := \int_0^t K(t, s)x(s)ds,
$$
where $K \in  C(D,\mathbb{R}^+)$, the set
of all positive function continuous on $D:= \{(t, s)\in
\mathbb{R}^2 : 0 \leq s \leq  t \leq  T\}$  and
\begin{equation}\label{defb*}
B^*= {\sup_{t\in [0,T]}}\int_0^t K(t, s)ds < \infty,
\end{equation}
$f$ and $g$ are continuous. The derivative $D^q$ is understood
here in the Riemann-Liouville sense. The non local condition
$$g(x)=\sum_{k=1}^{p}c_k x(t_k),$$ where $c_{k}$, $k=1,2,\dots p$,
are given constants and $0<t_1<t_2<\dots <t_p\leq T$. Let us recall
that such local  conditions  were first used by  Deng in
\cite{deng}. In his paper, Deng  indicated that using the
nonlocal condition $x(0) + g(x) = x_0$   to describe for instance,
the diffusion phenomenon of a small amount of gas in a transparent
tube can give  better result  than using the usual local Cauchy
Problem $x(0) = x_0$. Let's observe also that since Deng's paper,
such problem has also attracted several authors including
Aizicovici,  Byszewski, Ezzinbi,  Fan,  Liu,  Liang,
Lin,  Xiao,  Hern\'andez,  Lee, etc..(see for instance
\cite{aiz,karthi,bys,deng,hsiang,fan,liu,her,gaston1,gis1} and the
references therein).

 However, among the previous research on
nonlocal cauchy problems, few are concerned  with  mild solutions
of  fractional semilinear differential equations. Recently, in
\cite{jara}, the authors prove the  existence and uniqueness of a
mild solution for the semilinear initial value problem of
non-integer order when the linear part generates a strongly
continuous semigroup.  In \cite{gis2}, we considered the
fractional semilinear differential equation with nonlocal
conditions
\begin{equation}\label{e2.1}
\begin{gathered}
D^q x(t)=Ax(t)+t^{n}f(t,x(t),Bx(t)),\quad t\in [0,T],\quad
n\in \mathbb{Z}^{+}\\
x(0)=x_0+g(x)
\end{gathered}
\end{equation}
where $T$ is a positive real, $0<q<1$, $ A$ is the generator of a
$C_0$-semigroup  $(S(t))_{t\geq 0} $ on a Banach space
$\mathbb{X}$, $ Bx(t) := \int_0^t K(t, s)x(s)ds$, $K \in
C(D,\mathbb{R}^+)$  with $D := \{(t, s)\in  \mathbb{R}^2 : 0
\leq s \leq  t \leq  T\}$  and $
B^*= \displaystyle{\sup_{t\in [0,T]}}\int_0^t K(t, s)ds < \infty,
$
$f:\mathbb{R}\times \mathbb{X}\times \mathbb{X} \to \mathbb{X}$
is  a nonlinear function,  $g:\mathbb{C}([0,T],\mathbb{X}) \to D(A)$
is  continuous and  $0<q<1$. The derivative $D^q$ is understood
here in the Riemann-Liouville sense.

We used the Krasnoselkii and the contraction mapping principle to show
the existence and uniqueness of a  mild solution for a fractional
 semilinear differential equation with non local conditions.

In this paper, motivated by  \cite{ezzinbi2,hsiang}, we investigate
the existence and the uniqueness of a mild solution for the
fractional  semilinear differential equation  \eqref{e1.1},
assuming that $f$ is defined on
$[0,T]\times \mathbb{X}_\alpha\times \mathbb{X}_\alpha$ where
$\mathbb{X}_\alpha=D(A^\alpha)$, for some $0<\alpha<1$, the domain of
the fractional power of $A$.\par
The rest of this paper is organized as follows.
In section \ref{prelim} we give some known preliminary results
on the fractional powers of the generator of an analytic compact
semigroup. In Section \ref{main}, we study the existence and the
uniqueness of the mild solution for the  fractional  semilinear
differential equation  \eqref{e1.1}.


\section{Preliminaries\label{prelim}}

For the rest of this article, we set $I=[0,T]$. We denote by
$\mathbb{X}$ a Banach space with norm $\|\cdot\|$  and
$-A : D(A)\to  \mathbb{X} $ is the infinitesimal generator of a
compact analytic semigroup of
uniformly bounded linear operators $(S(t))_{t\geq 0}$. This means
that there exists $M>1$ such that
\begin{equation}\label{0.1}
\|S(t)\|\leq M
\end{equation}
 We assume without loss of generality that $0\in \rho(A)$.
This allows us to define the fractional power $A^\alpha$
for $0<\alpha<1$, as  a closed linear operator on its domain
$D(A^\alpha)$  with inverse $A^{-\alpha}$(see [8]).
We have the following  basic properties $A^\alpha$.

\begin{theorem}[{\cite[pp. 69-75]{pazy}}]\label{theo1}
\begin{enumerate}
\item $ \mathbb{X}_\alpha=D (A^\alpha)$  is a Banach space with
 the norm $\|x\|_\alpha := \|A^\alpha  x\|$ for $x\in D(A^\alpha)$.
\item $ S(t) : \mathbb{X} \to \mathbb{X}_\alpha$  for each $t > 0$.
\item $ A^\alpha S(t)x = S(t)A^\alpha x $ for each $x \in D(A^\alpha)$
  and $t \geq 0$.
\item  For every $t > 0$, $A^\alpha S(t)$  is bounded on $\mathbb{X}$
  and there exist $ M_\alpha > 0$ and $\delta > 0 $ such that
    \begin{equation}\label{0.2}
\|A^\alpha S(t)\|\leq {\frac{M_\alpha }{t^\alpha}}e^{-\delta t}
\end{equation}
\item $A^{-\alpha}$ is a bounded linear operator in $\mathbb{X}$
 with $D(A^\alpha) = \mathop{\rm Im}(A^{-\alpha})$.
\item  If $0 < \alpha \leq \beta$,  then
$ D(A^\beta)\hookrightarrow D(A^\alpha)$.
\end{enumerate}
\end{theorem}

\begin{remark} \label{rmk2.2} \rm
Observe as in \cite{hsiang} that by Theorem \ref{theo1} (ii) and (iii),
the restriction $S_\alpha(t)$ of $S(t)$ to $\mathbb{X}_\alpha$ is
exactly the part of $S(t)$ in $\mathbb{X}_\alpha$.
 Let $x \in \mathbb{X}_\alpha$. Since
$$
\|S(t)x\|_\alpha = \|A^\alpha S(t)x\| = \|S(t)A^\alpha x\|
 \leq \|S(t)\|\|A^\alpha x\| = \|S(t)\|\|x\|_\alpha,
$$
and as $t$ decreases to $0$
$$
\|S(t)x - x\|_\alpha=\|A^\alpha S(t)x - A^\alpha x\|
 = \|S(t)A^\alpha x - A^\alpha x\|\to  0,
$$
for all $x\in \mathbb{X}_\alpha$, it follows that
$(S(t))_{t\geq 0}$ is a family of strongly continuous semigroup on
$\mathbb{X}_\alpha$ and $\|S_\alpha (t)\|\leq \|S(t)\|$ for all
$t\geq 0$.
\end{remark}

 We have the the following  result from \cite{hsiang}.

\begin{lemma}\label{compact}
$ (S_\alpha(t))_{t\geq 0}$ is an immediately compact semigroup
in $\mathbb{X}_\alpha$, and hence it is immediately norm-continuous.
\end{lemma}


\begin{definition}[\cite{jara}] \rm
 A continuous function $x :  I\to \mathbb{X}$ satisfying the equation
\begin{equation}\label{defmild}
x(t)=S(t)(x_0-g(x))+
 {\frac{1}{\Gamma(q)}}\int_{0}^{t}(t-s)^{q-1} S(t-s)(f(s,x(s),Bx(s))\,ds
\end{equation}
for $t\in [0,T]$ is called a mild solution of the equation \eqref{e1.1}
\end{definition}

In the sequel, we  will also use $\|f\|_p$ to denote the $L^p$ norm
of $f$ whenever $f\in L^p(0, T)$ for some $p$ with $1\leq p<\infty$.
 We will set $\alpha \in (0,1)$
and we will denote by  $\mathcal{C}_\alpha$, the
Banach space $C([0, T], \mathbb{X}_\alpha)$ endowed with the supnorm given by
$$
\|x\|_\infty := \sup_{t\in I}\|x\|_\alpha ,\quad \text{for }
 x\in \mathcal{C}.
$$

\section{Main Results\label{main}}


We assume the following conditions:
\begin{itemize}
\item[(H1)] The function
$f : I \times \mathbb{X}_\alpha\times \mathbb{X}_\alpha \to \mathbb{X}$
is  continuous, and there exists a positive function
$\mu\in L_{\rm loc}^1(I, \mathbb{R}^+)$ such that
\begin{equation}\label{0.3}
\|f(t, x,y)\|\leq \mu(t),
\end{equation}

\item[(H2)] $g \in  C(\mathcal{C}_\alpha, \mathbb{X}_\alpha)$
is completely continuous and there exist $\lambda, \gamma > 0$ such that
$$
\|g(x)\|_\alpha \leq \lambda \|x\|_\infty + \gamma.
$$
\end{itemize}

\begin{theorem}\label{theo2}
 Suppose that assumptions {\rm (H1), (H2)} hold.
If $x_0\in \mathbb{X}_\alpha$ and
 \begin{equation}\label{0.4}
 M\lambda<{\frac{1}{2}}
 \end{equation}
then  \eqref{e1.1} has a mild solution on $[0, T]$.
\end{theorem}

\begin{proof}
We define  the function $F:\mathcal{C}_\alpha\to \mathcal{C}_\alpha$ by
$$
(Fx)(t)=S(t)(x_0-g(x))
+{\frac{1}{\Gamma(q)}}\int_0^t(t-s)^{q-1} S(t-s)f(s,x(s),Bx(s))\,ds,
$$
and we choose  $r$ such that
$$
r\geq 2\Big( {\frac{M_\alpha T^{q-\alpha}}{(1-\alpha)\Gamma(q)}}
\|\mu\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}
+M(\|x_0\|_\alpha+\gamma)\Big).
$$
 Let $B_r=\{x\in \mathcal{C}_\alpha:\|x\|_\infty\leq r\}$.
 Then we proceed in three steps.

\noindent\textbf{Step 1.} We show that $FB_r\subset B_r$.
Let $x\in B_r$. Then  for $t\in I$, we have
\begin{align*}
&\|(Fx)(t)\|_\alpha\\
&\leq  \|S(t)(x_0-g(x))\|_\alpha
 +{\frac{1}{\Gamma(q)}}\|\int_0^t(t-s)^{q-1} S(t-s)f(s,x(s),Bx(s))
  \,ds\|_\alpha\\
&\leq \|S(t)\|\|x_0-g(x)\|_\alpha
 + {\frac{1}{\Gamma(q)}}\int_0^t\|(t-s)^{q-1}A^\alpha S(t-s)f(s,x(s),
Bx(s))\|ds\\
&\leq \|S(t)\|(\|x_0\|_\alpha+ \lambda \|x\|_\infty+\gamma)
 + {\frac{T^{q-1}}{\Gamma(q)}}\int_0^t  \|A^\alpha S(t-s)\|
 \| f(s,x(s),Bx(s)) \|\,ds ,
\end{align*}
 which according to (\ref{0.1}), (\ref{0.2}), (\ref{0.3}) and
(\ref{0.4}) gives
\begin{align*}
&\|(Fx)(t)\|_\alpha\\
&\leq \|S(t)\|\left(\|x_0\|_\alpha+ \lambda \|x\|_\infty+\gamma\right)
+ {\frac{T^{q-1}}{\Gamma(q)}}\int_0^t  M_\alpha (t-s)^{-\alpha}
  e^{-\delta(t-s)}\mu(s)\,ds\\
&\leq \|S(t)\|\left(\|x_0\|_\alpha+ \lambda \|x\|_\infty+\gamma\right)
 + {\frac{M_\alpha T^{q-1}}{\Gamma(q)}}\int_0^t (t-s)^{-\alpha}
  \mu(s)\,ds\\
&\leq M\left(\|x_0\|_\alpha+ \lambda \|x\|_\infty+\gamma\right)
 + {\frac{M_\alpha T^{q-\alpha}}{(1-\alpha)\Gamma(q)}}
  \|\mu\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}
 \leq r
 \end{align*}
for $t\in I$.
 Hence, we deduce $ \|Fx\|_{\infty}\leq r$.

\noindent\textbf{Step 2.} We prove that $F$ is continuous.
Let $({x_n})$ be a sequence of  $B_r$ such that $x_n \to  x$ in $B_r$. Then
$$
f(s, x_n(s),Bx_n(s)) \to  f(s, x(s),Bx(s)), \quad n\to \infty
$$
because the function $f$ is continuous on
$I\times \mathbb{X}_\alpha\times \mathbb{X}_\alpha$.
Now, for $t\in I$, we have
\begin{align*}
&\|Fx_n-Fx\|_\alpha\\
&\leq \|S(t)(g(x_n) - g(x))\|_\alpha\\
&\quad + \big\|  {\frac{1}{\Gamma(q)}}\int_0^t (t-s)^{q-1}
S(t - s)\left(f(s, x_n(s),Bx_n(s)) - f(s, x(s),Bx(s))\right)\,ds\big\|_\alpha,
\end{align*}
which in view of (\ref{0.1}) and  (\ref{0.2}) gives
\begin{align*}
&\|Fx_n-Fx\|_\alpha\\
&\leq  \|S(t)\|\|g(x_n) - g(x)\|_\alpha\\
&\quad + {\frac{T^{q-1}}{\Gamma(q)}}\int_0^t
\|A^\alpha S(t - s)\|\|f(s, x_n(s),Bx_n(s)) - f(s, x(s),Bx(s))\|\,ds\\
&\leq  M\|g(x_n) - g(x)\|_\alpha\\
&\quad +  {\frac{M_\alpha T^{q-1}}{\Gamma(q)}}\int_0^t (t-s)^{-\alpha}
\|f(s, x_n(s),Bx_n(s)) - f(s, x(s),Bx(s))\|\,ds\\
&\leq  M\|g(x_n) - g(x)\|_\alpha\\
&\quad+  {\frac{M_\alpha T^{q-1}}{\Gamma(q)}}\int_0^t (t-s)^{-\alpha}
\|f(s, x_n(s),Bx_n(s)) - f(s, x(s),Bx(s))\|\,ds
\end{align*}
for  $t\in I$.  Therefore, using on the one hand the fact that
$$
\|f(s, x_n(s),Bx_n(s)) -  f(s, x(s),Bx(s))\|\leq 2\mu(s) \quad
\text{for } s \in  I,
$$
and for each $t\in  I$  since $f$ satisfies (H1)  and on the
other hand the fact that the function
$s\mapsto 2\mu(s) (t - s)^{-\alpha}$ is integrable on $I$,
by means of  the Lebesgue Dominated Convergence Theorem one proves that
$$
\int_0^t (t-s)^{-\alpha}
\|f(s, x_n(s),Bx_n(s)) - f(s, x(s),Bx(s))\|\,ds \to 0.
$$
Hence, since $g( x_n) \to  g(x)$ as $ n\to \infty$ because $g$
is completely continuous on  $\mathcal{C}_\alpha$,  it can easily been
shown that
$$
\lim_{n\to \infty}\|Fx_n-Fx\|_\infty = 0,
\quad\text{as }n\to \infty.
$$
In other words $F$ is continuous.

\noindent \textbf{Step 3.} We show that  $F$ is compact.
To this end, we use the Ascoli-Arzela's theorem.
We first prove that $\{(Fx)(t):x\in B_r\}$ is relatively compact
in $\mathbb{X}_\alpha$, for all $t\in I$. Obviously,
$\{(Fx)(0):x\in B_r\}$ is  compact.
Let $t\in (0,T]$. For each $h\in (0,t)$ and $x\in B_r$, we define
the operator $F_h$ by
\begin{align*}
(F_{h}x)(t)&= S(t)(x_0-g(x))+
 {\frac{1}{\Gamma(q)}}\int_{0}^{t-h}(t-s)^{q-1} S(t-s)f(s,x(s),Bx(s))\,ds\\
&= S(t)(x_0-g(x))+
 {\frac{S(h)}{\Gamma(q)}}\int_{0}^{t-h}(t-s)^{q-1} S(t-h-s)f(s,x(s),Bx(s))\,ds.
\end{align*}
Then the sets $\{(F_hx)(t):x\in B_r\}$ are relatively compact in
$\mathbb{X}_\alpha$ since by Lemma \ref{compact}, the operators
$S_\alpha(t)$, $t\geq 0$ are compact on $X_\alpha$. Moreover, using
(H1) and  (\ref{0.2}), we have
\begin{align*}
\|(Fx)(t)-(F_{h}x)(t)\|_\alpha
&\leq  {\frac{1}{\Gamma(q)}}\int_{t-h}^{t}(t-s)^{q-1}\|
S(t-s)f(s,x(s),Bx(s))\|_\alpha\,ds\\
&\leq  {\frac{T^{q-1}}{\Gamma(q)}}\int_{t-h}^{t}\|A^\alpha S(t-s)\|
\|f(s,x(s),Bx(s))\|\,ds\\
&\leq   {\frac{T^{q-1} M_\alpha}{\Gamma(q)}}\|\mu\|_{L^1_{\rm loc}
 (I,\mathbb{R}_+)}
\int_{t-h}^{t}(t-s)^{-\alpha}\,ds\\
&\leq  {\frac{T^{q-1} M_\alpha \|\mu\|_{L^1_{\rm loc}
(I,\mathbb{R}_+)}}{(1-\alpha)\Gamma(q)}}h^{1-\alpha}
\end{align*}
Therefore, we deduce that $\{(Fx)(t):x\in B_r\}$ is relatively  compact
in $\mathbb{X}_\alpha$  for all $t\in (0,T]$ and since it is compact
at $t=0$ we have the relatively compactness in $\mathbb{X}_\alpha$
for all $t\in I$.
Now, let us prove that $F(B_r)$ is equicontinuous. By the compactness
of the set $g(B_r)$, we can prove that the functions $Fx, x\in B_r$
are equicontinuous a $t=0$. For $0<t_2<t_1\leq T$, we have
\begin{align*}
&\|(Fx)(t_1)-(Fx)(t_2)\|_\alpha\\
&\leq  \|(S(t_1)-S(t_2))(x_0-g(x))\|_\alpha\\
&\quad +{\frac{1}{\Gamma(q)}}\|\int_{0}^{t_2}(t_1-s)^{q-1}
 \left(S(t_1-s)-S(t_2-s)\right)f(s,x(s),Bx(s)) \,ds\|_\alpha\\
&\quad + {\frac{1}{\Gamma(q)}}\|\int_{0}^{t_2}
  \left((t_1-s)^{q-1}-(t_2-s)^{q-1}\right) S(t_2-s)f(s,x(s),Bx(s))
  \,ds\|_\alpha\\
&\quad +  {\frac{1}{\Gamma(q)}}\|\int_{t_2}^{t_1}(t_1-s)^{q-1}
  S(t_1-s)f(s,x(s),Bx(s)) \,ds\|_\alpha\\
&\leq I_1+I_2+I_3+I_4
\end{align*}
 Where
\begin{gather*}
 I_1 = \|(S(t_1)-S(t_2))(x_0-g(x))\|_\alpha\\
 I_2 = {\frac{1}{\Gamma(q)}}\|\int_{0}^{t_2}(t_1-s)^{q-1}\left(S(t_1-s)-S(t_2-s)\right)f(s,x(s),Bx(s)) \,ds\|_\alpha\\
 I_3 =  {\frac{1}{\Gamma(q)}}\|\int_{0}^{t_2}\left((t_1-s)^{q-1}-(t_2-s)^{q-1}\right) S(t_2-s)f(s,x(s),Bx(s)) \,ds\|_\alpha\\
 I_4 =  {\frac{1}{\Gamma(q)}}\|\int_{t_2}^{t_1}(t_1-s)^{q-1} S(t_1-s)f(s,x(s),Bx(s)) \,ds\|_\alpha
\end{gather*}
 Actually, $I_1$, $I_2$, $I_3$ and $I_4$  tend to 0 independently
of $x\in B_r$ when $t_2\to t_1$. Indeed, let  $x\in B_r$
and $G= {\sup_{x\in \mathcal{C}_\alpha}}\|g(x)\|_\alpha$. We  have
\begin{align*}
 I_1&= \|(S(t_1)-S(t_2))(x_0-g(x))\|_\alpha\\
 &\leq  \|S_\alpha(t_1)-S_\alpha(t_2)\|_\alpha\|x_0-g(x)\|_\alpha\\
 &\leq \|S_\alpha(t_1)-S_\alpha(t_2)\|_\alpha\left(\|x_0\|_\alpha+ G\right)
\end{align*}
from which we deduce  that ${\lim_{t_2\to t_1}} I_1=0$ since by
Lemma \ref{compact} the function $t\mapsto \|S_\alpha(t)\|_\alpha$
is continuous for $t\in (0,T]$.
\begin{align*}
&I_2\\
&\leq   {\frac{1}{\Gamma(q)}}\int_{0}^{t_2}\|(t_1-s)^{q-1}
  \big(S(t_1-s)-S(t_2-s)\big)f(s,x(s),Bx(s)) \|_\alpha\,ds\\
&\leq  {\frac{T^{q-1}}{\Gamma(q)}}\int_{0}^{t_2}
\big\|\big[S\big( {\frac{t_1-t_2}{2}}+
  {\frac{t_1-s}{2}} \big)
 -S\big( {\frac{t_2-s}{2}}\big)\big]A^\alpha S
\big( {\frac{t_2-s}{2}}\big)f(s,x(s),Bx(s))\big\|\,ds\\
&\leq   {\frac{T^{q-1-\alpha}}{\Gamma(q)}}\|\mu\|_{L^1_{\rm loc}
(I,\mathbb{R}_+)}\int_{0}^{t_2}\|S\big( {\frac{t_1-t_2}{2}}+
  {\frac{t_1-s}{2}} \big)
 -S\big( {\frac{t_2-s}{2}}\big)\| \,ds.
 \end{align*}
Therefore, the continuity of the function $t\mapsto \|S(t)\|$ for
$t\in (0,T)$ allows us to conclude that
$ {\lim_{t_2\to t_1}} I_2=0$.

\begin{align*}
I_3&\leq  {\frac{1}{\Gamma(q)}}\int_{0}^{t_2}\|\left((t_2-s)^{q-1}-(t_1-s)^{q-1}\right) S(t_2-s)f(s,x(s),Bx(s)) \|_\alpha\,ds\\
 &\leq  {\frac{1}{\Gamma(q)}}\int_{0}^{t_2}\left|(t_2-s)^{q-1}-(t_1-s)^{q-1}\right| \|A^\alpha S(t_2-s)\|\|f(s,x(s),Bx(s)) \|\,ds\\
 &\leq  {\frac{1}{\Gamma(q)}}\int_{0}^{t_2}\left|(t_2-s)^{q-1}-(t_1-s)^{q-1}\right| (t_2-s)^{-\alpha}\mu(s)
 \,ds\\
 &\leq  {\frac{T^{-\alpha}}{\Gamma(q)}}\|\mu\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}\int_{0}^{t_2}\left|(t_2-s)^{q-1}-(t_1-s)^{q-1}\right|
 \,ds\\
 &\leq  {\frac{T^{-\alpha}}{q\Gamma(q)}}\|\mu\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}|t_1-t_2|^{q}.
 \end{align*}
Hence $ {\lim_{t_2\to t_1}} I_3=0$.

\begin{align*}
 I_4&\leq
  {\frac{1}{\Gamma(q)}}\int_{t_2}^{t_1}\|(t_1-s)^{q-1} S(t_1-s)f(s,x(s),Bx(s)) \|_\alpha\,ds\\
 &\leq
  {\frac{T^{q-1}}{\Gamma(q)}}\int_{t_2}^{t_1}\|A^\alpha S(t_1-s)\|\|f(s,x(s),Bx(s)) \|\,ds\\
 &\leq
  {\frac{M_\alpha T^{q-1}}{\Gamma(q)}}\int_{t_2}^{t_1}(t_1-s)^{-\alpha}\mu(s)\,ds\\
 &\leq
  {\frac{M_\alpha T^{q-1}}{(1-\alpha)\Gamma(q)}}
\|\mu\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}|t_1-t_2|^{1-\alpha}.
 \end{align*}
 Since $1-\alpha>0$, we deduce that $ {\lim_{t_2\to t_1}} I_4=0$.

In summary, we have proven that $F(B_r)$ is relatively compact,
for $t\in I$, $ \{Fx:x\in B_r\}$ is a family of equicontinuous functions.
Hence by the Arzela-Ascoli Theorem, $F$ is compact. By Schauder
fixed point theorem $F$ has a fixed point $x\in B_r$.
Consequently, \eqref{e1.1} has a mild solution.
 \end{proof}

Now we make the following assumptions.
\begin{itemize}
 \item[(H1')]  $f: I\times \mathbb{X}_\alpha\times
\mathbb{X}_\alpha\to \mathbb{X}$ is  continuous and
there exist functions
$\mu_1,\mu_2\in L^1_{\rm loc}(I,\mathbb{R}^+)$   such that
$$
\|f(t,x,u)-f(t,y,v)\|\leq \mu_1(t)\|x-y\|_\alpha+\mu_2(t)\|u-v\|_\alpha,
$$
 for all $t\in I$, $x,y,u,v\in\mathbb{X}_\alpha$.

\item[(H2')]  $g: \mathcal{C}_\alpha \to \mathbb{X}_\alpha$ is
continuous and  there exists a constant $b$  such that
$$
\|g(x)-g(y)\|_\alpha\leq b\|x-y\|_{\infty}, \quad \text{for all }x, y\in
\mathcal{C}_\alpha.
$$

\item[(H3)] The function $\Omega_{\alpha,q}:I\to \mathbb{R}_+$,
$0<\alpha,q<1$ defined by
$$
\Omega_{\alpha,q}= M b + {\frac{T^{q-1} M_\alpha
t^{1-\alpha}}{(1-\alpha)\Gamma(q)}}
\left(\|\mu_1\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}+
B^*\|\mu_2\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}\right)
$$
satisfies $0<\Omega_{\alpha,q}\leq\tau<1$, for all $t\in I$.
\end{itemize}


\begin{theorem}\label{theo1.1}
Assume that {\rm (H1'), (H2'), (H3)} hold.
If $x_0\in \mathbb{X}_\alpha$ then  \eqref{e1.1} has a unique mild
solution $x\in \mathcal{C}_\alpha$.
\end{theorem}

\begin{proof} Define  the function
$F:\mathcal{C}_\alpha\to \mathcal{C}_\alpha$ by
$$
(Fx)(t)= S(t)(x_0-g(x))
+ {\frac{1}{\Gamma(q)}}\int_0^t(t-s)^{q-1} S(t-s)f(s,x(s),Bx(s))\,ds.
$$
Note that $F$ is well defined on $\mathcal{C}_\alpha$.
Now take $t\in I$ and $x,y \in \mathcal{C}_\alpha$. We have
\begin{align*}
&\|(Fx)(t)-F(y)(t)\|_\alpha\\
&\leq \|S(t)\left(g(x)-g(y)\right)\|_\alpha\\
&\quad + {\frac{1}{\Gamma(q)}}\int_0^t(t-s)^{q-1} \|S(t-s)
\left(f(s,x(s),Bx(s))-f(s,y(s),By(s))\right)\|_\alpha\,ds\\
&\leq \|S(t)\|\|g(x)-g(y)\|_\alpha\\
&\quad + {\frac{T^{q-1}}{\Gamma(q)}}\int_0^t \| A^\alpha S(t-s)\|
\|f(s,x(s),Bx(s))-f(s,y(s),By(s))\| ds
\end{align*}
which according to (\ref{0.1}), (\ref{0.2}), (H1'), (H2')
and (\ref{defb*}) gives
\begin{align*}
&\|(Fx)(t)-F(y)(t)\|_\alpha\\
&\leq  M b\|x-y\|_\infty+ {\frac{T^{q-1} M_\alpha}{\Gamma(q)}}\int_0^t (t-s)^{-\alpha}\mu_1(s)
\|x(s)-y(s)\|_\alpha ds\\
&\quad + {\frac{T^{q-1} M_\alpha}{\Gamma(q)}}\int_0^t (t-s)^{-\alpha}\mu_2(s)
\|Bx(s)-By(s))\|_\alpha ds\\
&\leq  M b\|x-y\|_\infty\\
&\quad + {\frac{T^{q-1} M_\alpha}{\Gamma(q)}}\|\mu_1
 \|_{L^1_{\rm loc}(I,\mathbb{R}_+)}
\Big(\int_0^t (t-s)^{-\alpha}\,ds\Big)\|x(s)-y(s)\|_\infty\\
&\quad + {\frac{T^{q-1} M_\alpha}{\Gamma(q)}}\int_0^t
(t-s)^{-\alpha}\mu_2(s)\Big[
\int_{0}^sK(s,\sigma) \|A ^\alpha(x(\sigma)-y(\sigma))\| d \sigma \Big] ds\\
&\leq \Big( M b
+ {\frac{T^{q-1} M_\alpha t^{1-\alpha}}{(1-\alpha)\Gamma(q)}}
\|\mu_1\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}\Big)\|x-y\|_\infty\\
&\quad + {\frac{T^{q-1} M_\alpha B^* t^{1-\alpha}}{(1-\alpha)\Gamma(q)}}
\|\mu_2\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}\|x-y\|_\infty\\
&\leq \Big[ M b
+ {\frac{T^{q-1} M_\alpha t^{1-\alpha}}{(1-\alpha)\Gamma(q)}}
\left(\|\mu_1\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}+ B^*
\|\mu_2\|_{L^1_{\rm loc}(I,\mathbb{R}_+)}\right)\Big]\|x-y\|_\infty\\
&\leq\Omega_{\alpha,q}(t)\|x-y\|_\infty.
\end{align*}
So we get
$$
\|(Fx)(t)-F(y)(t)\|_\infty \leq \Omega_{\alpha,q}(t) \|x-y\|_\infty.
$$
Therefore, assumption (H3) allows us to conclude in view of
the contraction mapping principe that, $F$ has a unique fixed
point in $\mathcal{C}_\alpha$, and
$$
x(t)=S(t)(x_0-g(x))+
 {\frac{1}{\Gamma(q)}}\int_0^t(t-s)^{q-1} S(t-s)f(s,x(s),Bx(s))\,ds
$$
which is the mild solution of  \eqref{e1.1}.
\end{proof}

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\end{document}